Linear Regression Models - AP Statistics

1. What is the formula for calculating the correlation coefficient $r$?

   $r = \frac{\text{cov}(X, Y)}{\text{std}(X) \times \text{std}(Y)}$. Covariance divided by product of standard deviations.

2. Identify the y-intercept in $y = -2x + 6$.

   6. Y-intercept is the constant term when $x=0$.

3. What is the primary purpose of a regression model?

   Predict values of $y$. To estimate or forecast response variable values.

4. What does an $R^2$ value of 1 imply about the model?

   Perfect fit. All data points lie exactly on the regression line.

5. What is the interpretation of a slope of 0 in a regression model?

   No linear relationship. Zero slope means $y$ doesn't change as $x$ changes.

6. What is the formula for the least squares regression line?

   $y = b_0 + b_1x$. Standard form where $b_0$ is y-intercept and $b_1$ is slope.

7. Calculate the residual for $y_i = 8$ and $\bar{y}_i = 5$.

   3. Residual = observed - predicted = $8 - 5 = 3$.

8. What does the slope coefficient represent in a regression model?

   Change in $y$ per unit $x$. How much $y$ changes for each one-unit increase in $x$.

9. Find the predicted value of $y$ for $x=5$ in $y = 3x + 4$.

   19. Substitute $x=5$: $y = 3(5) + 4 = 19$.

10. State the formula to calculate residuals.

    $e_i = y_i - \bar{y}_i$. Difference between observed and predicted values.

11. What does a $p$-value less than 0.05 indicate?

    Statistically significant. Evidence to reject null hypothesis at 5% significance level.

12. What is the impact of high leverage points in regression?

    Influence regression line. Points far from mean $x$ have greater impact on line.

13. Identify the point where the regression line crosses the y-axis.

    Y-intercept. The value of $y$ when $x = 0$ (the constant term).

14. What does a residual of 0 indicate?

    Perfect prediction. Model predicted the exact observed value.

15. What does it mean if residuals show a pattern?

    Model is not appropriate. Patterns suggest non-linear relationship or other violations.

16. Find the regression line if $b_0=2$ and $b_1=4$.

    $y = 2 + 4x$. Substituting values into the standard form equation.

17. Identify the limitation of using $R^2$ alone to judge model fit.

    Does not show causation. High $R^2$ doesn't prove the variables are causally related.

18. What does extrapolation mean in regression?

    Predicting outside data range. Making predictions beyond the range of observed data.

19. State the independent variable in the equation $y = 5x + 3$.

    $x$. The predictor or explanatory variable in the equation.

20. What does a correlation coefficient of 0 imply?

    No linear relationship. Zero correlation means no linear association between variables.

21. Determine the effect of adding a constant to all $y$ values.

    Shifts intercept. Adding constant to all $y$ values shifts y-intercept up.

22. What does a positive slope indicate about the relationship between variables?

    Positive correlation. Positive slope means $y$ increases as $x$ increases.

23. State the condition when a linear model is appropriate.

    Linear relationship. Data points should follow a straight-line pattern.

24. Find the correlation coefficient if $R^2 = 0.81$.

    $\text{either } 0.9 \text{ or } -0.9$. $r = \pm\sqrt{R^2} = \pm\sqrt{0.81} = \pm 0.9$.

25. Find and correct: $y = 2x + 3$ is a nonlinear model.

    Linear model. $y = 2x + 3$ is linear since it has constant slope.

26. Identify the term for unexplained variability in regression.

    Error variance. Variability not explained by the regression model.

27. What is the significance of the $p$-value in regression?

    Tests hypothesis about slope. Tests if slope is significantly different from zero.

28. What is the effect of an outlier on a regression line?

    Influences the slope and intercept. Outliers can dramatically alter the regression line's position.

29. What is the purpose of a residual plot?

    Assess model fit. Shows patterns that indicate model assumptions violations.

30. What does $R^2$ represent in a linear regression model?

    Coefficient of determination. Measures proportion of variance in $y$ explained by the model.

31. Choose the term for residuals in regression analysis.

    Errors. Residuals are the differences between observed and predicted values.

32. Identify the dependent variable in regression analysis.

    Response variable. The variable being predicted or explained ($y$).

33. Identify the slope in the regression equation $y = 3 + 2x$.

    2. The slope is the coefficient of $x$ in the linear equation.

34. What is the interpretation of a negative slope?

    Negative correlation. Negative slope means $y$ decreases as $x$ increases.

35. Find the $R^2$ if the correlation $r$ is 0.8.

    0.64. $R^2 = r^2$, so $(0.8)^2 = 0.64$.

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